Editing Hyperbolic coordinates (section)

==Alternative quadrant metric==
Since ''HP'' carries the [[metric space]] structure of the [[Poincaré half-plane model]] of [[hyperbolic geometry]], the bijective correspondence 
<math>Q \leftrightarrow HP</math> brings this structure to ''Q''. It can be grasped using the notion of [[hyperbolic motion]]s. Since [[geodesic]]s in ''HP'' are semicircles with centers on the boundary, the geodesics in ''Q'' are obtained from the correspondence and turn out to be [[Line (mathematics)#Ray|rays]] from the origin or [[petal]]-shaped [[curve]]s leaving and re-entering the origin. And the hyperbolic motion of ''HP'' given by a left-right shift corresponds to a [[squeeze mapping]] applied to ''Q''.

Since [[hyperbola]]s in ''Q'' correspond to lines parallel to the boundary of ''HP'', they are [[horocycle]]s in the metric geometry of ''Q''.

If one only considers the [[Euclidean topology]] of the plane and the topology inherited by ''Q'', then the lines bounding ''Q'' seem close to ''Q''. Insight from the metric space ''HP'' shows that the [[open set]] ''Q'' has only the [[origin (mathematics)|origin]] as boundary when viewed through the correspondence. Indeed, consider rays from the origin in ''Q'', and their images, vertical rays from the boundary ''R'' of ''HP''. Any point in ''HP'' is an infinite distance from the point ''p'' at the foot of the perpendicular to ''R'', but a sequence of points on this perpendicular may tend in the direction of ''p''. The corresponding sequence in ''Q'' tends along a ray toward the origin. The old Euclidean boundary of ''Q'' is no longer relevant.